Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T12:57:25.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Compatible finite element methods for geophysical fluid dynamics

Part of: Geophysics Numerical problems in dynamical systems Geophysics (general) Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  11 May 2023

Colin J. Cotter Open the ORCID record for Colin J. Cotter [Opens in a new window]
Colin J. Cotter*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

MSC classification

Primary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65P10: Hamiltonian systems including symplectic integrators 65Z05: Applications to physics 86A05: Hydrology, hydrography, oceanography 86A10: Meteorology and atmospheric physics
Type
Research Article
Information
Acta Numerica , Volume 32 , May 2023 , pp. 291 - 393
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

References

Adams, S. V., Ford, R. W., Hambley, M., Hobson, J. M., Kavčič, I., Maynard, C. M., Melvin, T., Müller, E. H., Mullerworth, S., Porter, A. R. et al. (2019), LFRic: Meeting the challenges of scalability and performance portability in weather and climate models, J. Parallel Distrib. Comput. 132, 383–396.CrossRefGoogle Scholar
Arakawa, A. (1966), Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow, Part I, J. Comput. Phys. 1, 119–143.CrossRefGoogle Scholar
Arakawa, A. and Hsu, Y.-J. G. (1990), Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations, Mon Weather Rev. 118, 1960–1969.2.0.CO;2>CrossRefGoogle Scholar
Arakawa, A. and Lamb, V. R. (1977), Computational design of the basic dynamical processes of the UCLA general circulation model, Methods Comput. Phys. 17, 173–265.Google Scholar
Arakawa, A. and Lamb, V. R. (1981), A potential enstrophy and energy conserving scheme for the shallow water equations, Mon Weather Rev. 109, 18–36.2.0.CO;2>CrossRefGoogle Scholar
Arbogast, T. and Correa, M. R. (2016), Two families of H(div) mixed finite elements on quadrilaterals of minimal dimension, SIAM J. Numer. Anal. 54, 3332–3356.CrossRefGoogle Scholar
Arnold, D. N. (2018), Finite Element Exterior Calculus, SIAM.CrossRefGoogle Scholar
Arnold, D. N. and Awanou, G. (2014), Finite element differential forms on cubical meshes, Math. Comp. 83, 1551–1570.CrossRefGoogle Scholar
Arnold, D. N., Boffi, D. and Falk, R. S. (2005), Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. 42, 2429–2451.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006), Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2010), Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc. 47, 281–354.CrossRefGoogle Scholar
Arnold, V. (1966), Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann Inst. Fourier 16, 319–361.CrossRefGoogle Scholar
Badin, G., Oliver, M. and Vasylkevych, S. (2018), Geometric Lagrangian averaged Euler–Boussinesq and primitive equations, J. Phys. A 51, 455501.CrossRefGoogle Scholar
Bauer, W. and Cotter, C. J. (2018), Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions, J. Comput. Phys. 373, 171–187.CrossRefGoogle Scholar
Bauer, W. and Gay-Balmaz, F. (2017), Variational integrators for anelastic and pseudo-incompressible flows, J. Geom. Mech. 11, 511–537.CrossRefGoogle Scholar
Bauer, W. and Gay-Balmaz, F. (2019), Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations, J. Comput. Dyn. 6, 1–37.Google Scholar
Bendall, T. M. and Wimmer, G. A. (2023), Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart–Thomas finite elements, J. Comput. Phys. 474, 111834.CrossRefGoogle Scholar
Bendall, T. M., Cotter, C. J. and Shipton, J. (2019), The ‘recovered space’ advection scheme for lowest-order compatible finite element methods, J. Comput. Phys. 390, 342–358.CrossRefGoogle Scholar
Bendall, T. M., Gibson, T. H., Shipton, J., Cotter, C. J. and Shipway, B. (2020), A compatible finite-element discretisation for the moist compressible Euler equations, Quart. J. Royal Meteorol. Soc. 146, 3187–3205.CrossRefGoogle Scholar
Bendall, T. M., Wood, N., Thuburn, J. and Cotter, C. J. (2022), A solution to the trilemma of the moist Charney–Phillips staggering, Quart. J. Royal Meteorol. Soc. 149, 262–276.CrossRefGoogle Scholar
Bercea, G.-T., McRae, A. T. T., Ham, D. A., Mitchell, L., Rathgeber, F., Nardi, L., Luporini, F. and Kelly, P. H. J. (2016), A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake, Geosci Model Dev. 9, 3803–3815.CrossRefGoogle Scholar
Bernard, P.-E., Remacle, J.-F., Comblen, R., Legat, V. and Hillewaert, K. (2009), High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations, J. Comput. Phys. 228, 6514–6535.CrossRefGoogle Scholar
Bernsen, E., Bokhove, O. and van der Vegt, J. J. W. (2006), A (dis) continuous finite element model for generalized 2D vorticity dynamics, J. Comput. Phys. 211, 719–747.CrossRefGoogle Scholar
Betteridge, J. D., Cotter, C. J., Gibson, T. H., Griffith, M. J., Melvin, T. and Müller, E. H. (2022), Hybridised multigrid preconditioners for a compatible finite element dynamical core. Available at arXiv:2210.11797.Google Scholar
Bochev, P. B. and Ridzal, D. (2009), Rehabilitation of the lowest-order Raviart–Thomas element on quadrilateral grids, SIAM J. Numer. Anal. 47, 487–507.CrossRefGoogle Scholar
Boffi, D., Brezzi, F. and Fortin, M. (2013), Mixed Finite Element Methods and Applications, Vol. 44 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Bou-Rabee, N. and Marsden, J. E. (2009), Hamilton–Pontryagin integrators on Lie groups, Part I: Introduction and structure-preserving properties, Found. Comput. Math. 9, 197–219.CrossRefGoogle Scholar
Brecht, R., Bauer, W., Bihlo, A., Gay-Balmaz, F. and MacLachlan, S. (2019), Variational integrator for the rotating shallow-water equations on the sphere, Quart. J. Royal Meteorol. Soc. 145, 1070–1088.CrossRefGoogle Scholar
Brezzi, F., Marini, L. D. and Süli, E. (2004), Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci. 14, 1893–1903.CrossRefGoogle Scholar
Brooks, A. N. and Hughes, T. J. R. (1982), Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg 32, 199–259.CrossRefGoogle Scholar
Burman, E. and Hansbo, P. (2004), Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems, Comput Methods Appl. Mech. Engrg 193, 1437–1453.CrossRefGoogle Scholar
Bush, M., Allen, T., Bain, C., Boutle, I., Edwards, J., Finnenkoetter, A., Franklin, C., Hanley, K., Lean, H., Lock, A. et al. (2020), The first Met Office Unified Model–JULES regional atmosphere and land configuration, RAL1, Geosci Model Dev. 13, 1999–2029.CrossRefGoogle Scholar
Chen, Q., Gunzburger, M. and Ringler, T. (2012), A scale-aware anticipated potential vorticity method: On variable-resolution meshes, Mon Weather Rev. 140, 3127–3133.CrossRefGoogle Scholar
Christiansen, S. H. (2011), On the linearization of Regge calculus, Numer Math. 119, 613–640.CrossRefGoogle Scholar
Christiansen, S. H., Gopalakrishnan, J., Guzmán, J. and Hu, K. (2020), A discrete elasticity complex on three-dimensional Alfeld splits. Available at arXiv:2009.07744.Google Scholar
Cockburn, B. and Gopalakrishnan, J. (2004), A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42, 283–301.CrossRefGoogle Scholar
Cohen, D. and Hairer, E. (2011), Linear energy-preserving integrators for Poisson systems, BIT Numer. Math. 51, 91–101.CrossRefGoogle Scholar
Cotter, C. J. and Cullen, M. J. P. (2019), Particle relabelling symmetries and Noether’s theorem for vertical slice models, J. Geom. Mech. 11, 139.CrossRefGoogle Scholar
Cotter, C. J. and Holm, D. D. (2013a), On Noether’s theorem for the Euler–Poincaré equation on the diffeomorphism group with advected quantities, Found. Comput. Math. 13, 457–477.CrossRefGoogle Scholar
Cotter, C. J. and Holm, D. D. (2013b), A variational formulation of vertical slice models, Proc Royal Soc. A: Math. Phys. Engrg Sci. 469, 20120678.CrossRefGoogle Scholar
Cotter, C. J. and Holm, D. D. (2014), Variational formulations of sound-proof models, Quart. J. Royal Meteorol. Soc. 140, 1966–1973.CrossRefGoogle Scholar
Cotter, C. J. and Kirby, R. C. (2016), Mixed finite elements for global tide models, Numer Math. 133, 255–277.CrossRefGoogle ScholarPubMed
Cotter, C. J. and Kuzmin, D. (2016), Embedded discontinuous Galerkin transport schemes with localised limiters, J. Comput. Phys. 311, 363–373.CrossRefGoogle Scholar
Cotter, C. J. and Shipton, J. (2012), Mixed finite elements for numerical weather prediction, J. Comput. Phys. 231, 7076–7091.CrossRefGoogle Scholar
Cotter, C. J. and Shipton, J. (2022), A compatible finite element discretisation for the nonhydrostatic vertical slice equations. Available at arXiv:2210.07861.Google Scholar
Cotter, C. J. and Thuburn, J. (2014), A finite element exterior calculus framework for the rotating shallow-water equations, J. Comput. Phys. 257, 1506–1526.CrossRefGoogle Scholar
Cotter, C. J., Graber, P. J. and Kirby, R. C. (2018), Mixed finite elements for global tide models with nonlinear damping, Numer Math. 140, 963–991.CrossRefGoogle Scholar
Cotter, C. J., Kirby, R. C. and Morris, H. (2022), Weighted-norm preconditioners for a multi-layer tide model. Available at arXiv:2207.02116 (to appear in SIAM J. Sci. Comput.).Google Scholar
Danilov, S. (2010), On utility of triangular C-grid type discretization for numerical modeling of large-scale ocean flows, Ocean Dyn. 60, 1361–1369.CrossRefGoogle Scholar
Danilov, S. and Kutsenko, A. (2019), On the geometric origin of spurious waves in finite-volume discretizations of shallow water equations on triangular meshes, J. Comput. Phys. 398, 108891.CrossRefGoogle Scholar
Dennis, J. M., Edwards, J., Evans, K. J., Guba, O., Lauritzen, P. H., Mirin, A. A., St-Cyr, A., Taylor, M. A. and Worley, P. H. (2012), CAM-se: A scalable spectral element dynamical core for the Community Atmosphere Model, Internat. J. High Perform. Comput. Appl. 26, 74–89.CrossRefGoogle Scholar
Desbrun, M., Gawlik, E. S., Gay-Balmaz, F. and Zeitlin, V. (2014), Variational discretization for rotating stratified fluids, Discrete Contin Dyn. Syst. 34, 477–509.Google Scholar
Dubinkina, S. (2018), Relevance of conservative numerical schemes for an ensemble Kalman filter, Quart. J. Royal Meteorol. Soc. 144, 468–477.CrossRefGoogle Scholar
Dubinkina, S. and Frank, J. (2007), Statistical mechanics of Arakawa’s discretizations, J. Comput. Phys. 227, 1286–1305.CrossRefGoogle Scholar
Dubos, T., Dubey, S., Tort, M., Mittal, R., Meurdesoif, Y. and Hourdin, F. (2015), DYNAMICO-1.0: An icosahedral hydrostatic dynamical core designed for consistency and versatility, Geosci Model Dev. 8, 3131–3150.CrossRefGoogle Scholar
Eldred, C. and Bauer, W. (2022), An interpretation of TRiSK-type schemes from a discrete exterior calculus perspective. Available at arXiv:2210.07476.Google Scholar
Eldred, C. and Le Roux, D. Y. (2018), Dispersion analysis of compatible Galerkin schemes for the 1D shallow water model, J. Comput. Phys. 371, 779–800.CrossRefGoogle Scholar
Eldred, C. and Le Roux, D. Y. (2019), Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models, J. Comput. Phys. 387, 539–568.CrossRefGoogle Scholar
Eldred, C. and Randall, D. (2017), Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods, Part 1: Derivation and properties, Geosci Model Dev. 10, 791–810.CrossRefGoogle Scholar
Eldred, C., Dubos, T. and Kritsikis, E. (2019), A quasi-Hamiltonian discretization of the thermal shallow water equations, J. Comput. Phys. 379, 1–31.CrossRefGoogle Scholar
Evans, L. C. (2022), Partial Differential Equations , Vol. 19 of Graduate Studies in Mathematics, AMS. Google Scholar
Falk, R. S. and Neilan, M. (2013), Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal. 51, 1308–1326.CrossRefGoogle Scholar
Falk, R. S., Gatto, P. and Monk, P. (2011), Hexahedral H(div) and H(curl) finite elements, ESAIM Math. Model. Numer. Anal. 45, 115–143.CrossRefGoogle Scholar
Gassmann, A. (2013), A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency, Quart. J. Royal Meteorol. Soc. 139, 152–175.CrossRefGoogle Scholar
Gassmann, A. and Herzog, H.-J. (2008), Towards a consistent numerical compressible non-hydrostatic model using generalized Hamiltonian tools, Quart. J. Royal Meteorol. Soc. 134, 1597–1613.CrossRefGoogle Scholar
Gawlik, E. S. and Gay-Balmaz, F. (2021a), A structure-preserving finite element method for compressible ideal and resistive magnetohydrodynamics, J. Plasma Phys. 87, 835870501.CrossRefGoogle Scholar
Gawlik, E. S. and Gay-Balmaz, F. (2021b), A variational finite element discretization of compressible flow, Found Comput. Math. 21, 961–1001.CrossRefGoogle Scholar
Gay-Balmaz, F. (2019), A variational derivation of the thermodynamics of a moist atmosphere with rain process and its pseudoincompressible approximation, Geophys Astrophys. Fluid Dyn. 113, 428–465.CrossRefGoogle Scholar
Gay-Balmaz, F. and Ratiu, T. S. (2009), The geometric structure of complex fluids, Adv. Appl. Math. 42, 176–275.CrossRefGoogle Scholar
Georgoulis, E. H. and Pryer, T. (2018), Recovered finite element methods, Comput Methods Appl. Mech. Engrg 332, 303–324.CrossRefGoogle Scholar
Gibson, T. H. (2019), Hybridizable compatible finite element discretizations for numerical weather prediction: Implementation and analysis. PhD thesis, Imperial College London.Google Scholar
Gibson, T. H., McRae, A. T. T., Cotter, C. J., Mitchell, L. and Ham, D. A. (2019), Compatible Finite Element Methods for Geophysical Flows: Automation and Implementation Using Firedrake, Springer.CrossRefGoogle Scholar
Gibson, T. H., Mitchell, L., Ham, D. A. and Cotter, C. J. (2020), SLATE: Extending Firedrake’s domain-specific abstraction to hybridized solvers for geoscience and beyond, Geosci Model Dev. 13, 735–761.CrossRefGoogle Scholar
Gillette, A. and Kloefkorn, T. (2019), Trimmed serendipity finite element differential forms, Math. Comp. 88, 583–606.CrossRefGoogle Scholar
Giraldo, F. X., Kelly, J. F. and Constantinescu, E. M. (2013), Implicit–explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA), SIAM J. Sci. Comput. 35, B1162–B1194.CrossRefGoogle Scholar
Gopalakrishnan, J. (2003), A Schwarz preconditioner for a hybridized mixed method, Comput Methods Appl. Math. 3, 116–134.CrossRefGoogle Scholar
Gopalakrishnan, J. and Tan, S. (2009), A convergent multigrid cycle for the hybridized mixed method, Numer Linear Algebra Appl. 16, 689–714.CrossRefGoogle Scholar
Guzmán, J., Shu, C.-W. and Sequeira, F. A. (2017), H(div) conforming and DG methods for incompressible Euler’s equations, IMA J. Numer. Anal. 37, 1733–1771.Google Scholar
Hairer, E. (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math. 5, 73–84.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2003), Geometric numerical integration illustrated by the Störmer–Verlet method, Acta Numer. 12, 399–450.CrossRefGoogle Scholar
Hairer, E., Wanner, G. and Lubich, C. (2006), Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, Springer.Google Scholar
Heumann, H. and Hiptmair, R. (2011), Eulerian and semi-Lagrangian methods for convection–diffusion for differential forms, Discrete Cont Dyn. Syst. 29, 1471–1495.CrossRefGoogle Scholar
Hirani, A. N. (2003), Discrete Exterior Calculus, California Institute of Technology.Google Scholar
Hollingsworth, A., Kållberg, P., Renner, V. and Burridge, D. M. (1983), An internal symmetric computational instability, Quart. J. Royal Meteorol. Soc. 109, 417–428.CrossRefGoogle Scholar
Holm, D. D. (2002), Euler–Poincaré dynamics of perfect complex fluids, in Geometry , Mechanics, and Dynamics (Newton, P. et al., eds), Springer, pp. 169–180.Google Scholar
Holm, D. D. and Zeitlin, V. (1998), Hamilton’s principle for quasigeostrophic motion, Phys. Fluids 10, 800–806.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E. and Ratiu, T. S. (1998), The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv Math. 137, 1–81.CrossRefGoogle Scholar
Holst, M. and Stern, A. (2012), Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces, Found. Comput. Math. 12, 263–293.CrossRefGoogle Scholar
Hu, K., Zhang, Q. and Zhang, Z. (2022), A family of finite element Stokes complexes in three dimensions, SIAM J. Numer. Anal. 60, 222–243.CrossRefGoogle Scholar
Kent, J., Melvin, T. and Wimmer, G. A. (2023), A mixed finite-element discretisation of the shallow-water equations, Geosci Model Dev. 16, 1265–1276.CrossRefGoogle Scholar
Ketefian, G. S. and Jacobson, M. Z. (2009), A mass, energy, vorticity, and potential enstrophy conserving lateral fluid–land boundary scheme for the shallow water equations, J. Comput. Phys. 228, 1–32.CrossRefGoogle Scholar
Kirby, R. C. and Kernell, T. (2021), Preconditioning mixed finite elements for tide models, Comput. Math. Appl. 82, 212–227.CrossRefGoogle Scholar
Kuzmin, D. (2013), Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis, Internat. J. Numer. Methods Fluids 71, 1178–1190.CrossRefGoogle Scholar
Le Roux, D. Y. (2012), Spurious inertial oscillations in shallow-water models, J. Comput. Phys. 231, 7959–7987.CrossRefGoogle Scholar
Le Roux, D. Y. and Pouliot, B. (2008), Analysis of numerically induced oscillations in two-dimensional finite-element shallow-water models, Part II: Free planetary waves, SIAM J. Sci. Comput. 30, 1971–1991.CrossRefGoogle Scholar
Le Roux, D. Y., Rostand, V. and Pouliot, B. (2007), Analysis of numerically induced oscillations in 2D finite-element shallow-water models, Part I: Inertia-gravity waves, SIAM J. Sci. Comput. 29, 331–360.CrossRefGoogle Scholar
Le Roux, D. Y., Staniforth, A. and Lin, C. A. (1998), Finite elements for shallow-water equation ocean models, Mon Weather Rev. 126, 1931–1951.2.0.CO;2>CrossRefGoogle Scholar
Lee, D. (2021), Petrov–Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems, Comput. Math. Appl. 89, 68–77.CrossRefGoogle Scholar
Lee, D. and Palha, A. (2018), A mixed mimetic spectral element model of the rotating shallow water equations on the cubed sphere, J. Comput. Phys. 375, 240–262.CrossRefGoogle Scholar
Lee, D. and Palha, A. (2021), Exact spatial and temporal balance of energy exchanges within a horizontally explicit/vertically implicit non-hydrostatic atmosphere, J. Comput. Phys. 440, 110432.CrossRefGoogle Scholar
Lee, D., Palha, A. and Gerritsma, M. (2018), Discrete conservation properties for shallow water flows using mixed mimetic spectral elements, J. Comput. Phys. 357, 282–304.CrossRefGoogle Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics , Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Liu, J.-G. and W, E (2001), Simple finite element method in vorticity formulation for incompressible flows, Math. Comp. 70, 579–593.CrossRefGoogle Scholar
Liu, J.-G. and Shu, C.-W. (2000), A high-order discontinuous Galerkin method for 2D incompressible flows, J. Comput. Phys. 160, 577–596.CrossRefGoogle Scholar
Logg, A., Mardal, K.-A. and Wells, G. (2012), Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Vol. 84 of Lecture Notes in Computational Science and Engineering, Springer.CrossRefGoogle Scholar
Marsden, J. and Weinstein, A. (1983), Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7, 305–323.CrossRefGoogle Scholar
Marsden, J. E. and West, M. (2001), Discrete mechanics and variational integrators, Acta Numer. 10, 357–514.CrossRefGoogle Scholar
Maynard, C., Melvin, T. and Müller, E. H. (2020), Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model, Quart. J. Royal Meteorol. Soc. 146, 3917–3936.CrossRefGoogle Scholar
McRae, A. T. T. and Cotter, C. J. (2014), Energy-and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements, Quart. J. Royal Meteorol. Soc. 140, 2223–2234.CrossRefGoogle Scholar
McRae, A. T. T., Bercea, G.-T., Mitchell, L., Ham, D. A. and Cotter, C. J. (2016), Automated generation and symbolic manipulation of tensor product finite elements, SIAM J. Sci. Comput. 38, S25–S47.CrossRefGoogle Scholar
Melvin, T. and Thuburn, J. (2017), Wave dispersion properties of compound finite elements, J. Comput. Phys. 338, 68–90.CrossRefGoogle Scholar
Melvin, T., Benacchio, T., Shipway, B., Wood, N., Thuburn, J. and Cotter, C. (2019), A mixed finite-element, finite-volume, semi-implicit discretization for atmospheric dynamics: Cartesian geometry, Quart. J. Royal Meteorol. Soc. 145, 2835–2853.CrossRefGoogle Scholar
Melvin, T., Benacchio, T., Thuburn, J. and Cotter, C. (2018), Choice of function spaces for thermodynamic variables in mixed finite-element methods, Quart. J. Royal Meteorol. Soc. 144, 900–916.CrossRefGoogle Scholar
Melvin, T., Staniforth, A. and Cotter, C. (2014), A two-dimensional mixed finite-element pair on rectangles, Quart. J. Royal Meteorol. Soc. 140, 930–942.CrossRefGoogle Scholar
Mitchell, L. and Müller, E. H. (2016), High level implementation of geometric multigrid solvers for finite element problems: Applications in atmospheric modelling, J. Comput. Phys. 327, 1–18.CrossRefGoogle Scholar
Morrison, P. J. (1982), Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, Vol. 88 of AIP Conference Proceedings, American Institute of Physics, pp. 13–46.Google Scholar
Natale, A. and Cotter, C. J. (2017), Scale-selective dissipation in energy-conserving finite-element schemes for two-dimensional turbulence, Quart. J. Royal Meteorol. Soc. 143, 1734–1745.CrossRefGoogle Scholar
Natale, A. and Cotter, C. J. (2018), A variational finite-element discretization approach for perfect incompressible fluids, IMA J. Numer. Anal. 38, 1388–1419.CrossRefGoogle Scholar
Natale, A., Shipton, J. and Cotter, C. J. (2016), Compatible finite element spaces for geophysical fluid dynamics, Dyn Statist. Climate Syst. 1, dzw005.Google Scholar
Neilan, M. (2020), The Stokes complex: A review of exactly divergence-free finite element pairs for incompressible flows, in 75 Years of Mathematics of Computation, Vol. 754 of Contemporary Mathematics, AMS, pp. 141–158.CrossRefGoogle Scholar
Pavlov, D., Mullen, P., Tong, Y., Kanso, E., Marsden, J. E. and Desbrun, M. (2011), Structure-preserving discretization of incompressible fluids, Phys. D 240, 443–458.CrossRefGoogle Scholar
Rathgeber, F., Ham, D. A., Mitchell, L., Lange, M., Luporini, F., McRae, A. T. T., Bercea, G.-T., Markall, G. R. and Kelly, P. H. (2016), Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software 43, 1–27.CrossRefGoogle Scholar
Ringler, T. D., Thuburn, J., Klemp, J. B. and Skamarock, W. C. (2010), A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids, J. Comput. Phys. 229, 3065–3090.CrossRefGoogle Scholar
Ripa, P. (1993), Conservation laws for primitive equations models with inhomogeneous layers, Geophys Astrophys. Fluid Dyn. 70, 85–111.CrossRefGoogle Scholar
Rognes, M. E., Ham, D. A., Cotter, C. J. and McRae, A. T. T. (2013), Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2, Geosci Model Dev. 6, 2099–2119.CrossRefGoogle Scholar
Rognes, M. E., Kirby, R. C. and Logg, A. (2010), Efficient assembly of H(div) and H(curl) conforming finite elements, SIAM J. Sci. Comput. 31, 4130–4151.CrossRefGoogle Scholar
Rostand, V. and Le Roux, D. Y. (2008), Raviart–Thomas and Brezzi–Douglas–Marini finite-element approximations of the shallow-water equations, Internat. J. Numer. Methods Fluids 57, 951–976.CrossRefGoogle Scholar
Sadourny, R. (1972), Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Mon Weather Rev. 100, 136–144.2.3.CO;2>CrossRefGoogle Scholar
Sadourny, R. (1975), The dynamics of finite-difference models of the shallow-water equations, J. Atmos. Sci. 32, 680–689.2.0.CO;2>CrossRefGoogle Scholar
Sadourny, R. and Basdevant, C. (1985), Parameterization of subgrid scale barotropic and baroclinic eddies in quasi-geostrophic models: Anticipated potential vorticity method, J. Atmos. Sci. 42, 1353–1363.2.0.CO;2>CrossRefGoogle Scholar
Sadourny, R., Arakawa, A. and Mintz, Y. (1968), Integration of the nondivergent barotropic vorticity equation with an icosahedral–hexagonal grid for the sphere, Mon Weather Rev. 96, 351–356.2.0.CO;2>CrossRefGoogle Scholar
Salmon, R. (1983), Practical use of Hamilton’s principle, J. Fluid Mech. 132, 431–444.CrossRefGoogle Scholar
Salmon, R. (1998), Lectures on Geophysical Fluid Dynamics, Oxford University Press.CrossRefGoogle Scholar
Sanz-Serna, J. M. (1992), Symplectic integrators for Hamiltonian problems: An overview, Acta Numer. 1, 243–286.CrossRefGoogle Scholar
Shepherd, T. G. (1990), Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics, Adv Geophys. 32, 287–338.CrossRefGoogle Scholar
Shipton, J., Gibson, T. H. and Cotter, C. J. (2018), Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere, J. Comput. Phys. 375, 1121–1137.CrossRefGoogle Scholar
Skamarock, W. C., Klemp, J. B., Duda, M. G., Fowler, L. D., Park, S.-H. and Ringler, T. D. (2012), A multiscale nonhydrostatic atmospheric model using centroidal Voronoi tesselations and C-grid staggering, Mon Weather Rev. 140, 3090–3105.CrossRefGoogle Scholar
Sommer, M. and Névir, P. (2009), A conservative scheme for the shallow-water system on a staggered geodesic grid based on a Nambu representation, Quart. J. Royal Meteorol. Soc. 135, 485–494.CrossRefGoogle Scholar
Staniforth, A. and Thuburn, J. (2012), Horizontal grids for global weather and climate prediction models: A review, Quart J. Royal Meteorol. Soc. 138, 1–26.CrossRefGoogle Scholar
Staniforth, A., Melvin, T. and Cotter, C. (2013), Analysis of a mixed finite-element pair proposed for an atmospheric dynamical core, Quart. J. Royal Meteorol. Soc. 139, 1239–1254.CrossRefGoogle Scholar
Tai, X.-C. and Winther, R. (2006), A discrete de Rham complex with enhanced smoothness, Calcolo 43, 287–306.CrossRefGoogle Scholar
Tezduyar, T. E. (1989), Finite element formulation for the vorticity-stream function form of the incompressible Euler equations on multiply-connected domains, Comput Methods Appl. Mech. Engrg 73, 331–339.CrossRefGoogle Scholar
Tezduyar, T. E., Glowinski, R. and Liou, J. (1988), Petrov–Galerkin methods on multiply connected domains for the vorticity–stream function formulation of the incompressible Navier–Stokes equations, Internat. J. Numer. Methods Fluids 8, 1269–1290.CrossRefGoogle Scholar
Thuburn, J. (2008), Numerical wave propagation on the hexagonal C-grid, J. Comput. Phys. 227, 5836–5858.CrossRefGoogle Scholar
Thuburn, J. and Cotter, C. J. (2012), A framework for mimetic discretization of the rotating shallow-water equations on arbitrary polygonal grids, SIAM J. Sci. Comput. 34, B203–B225.CrossRefGoogle Scholar
Thuburn, J. and Cotter, C. J. (2015), A primal–dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, J. Comput. Phys. 290, 274–297.CrossRefGoogle Scholar
Thuburn, J., Cotter, C. J. and Dubos, T. (2014a), A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: Comparison of hexagonal–icosahedral and cubed-sphere grids, Geosci Model Dev. 7, 909–929.CrossRefGoogle Scholar
Thuburn, J., Kent, J. and Wood, N. (2014b), Cascades, backscatter and conservation in numerical models of two-dimensional turbulence, Quart. J. Royal Meteorol. Soc. 140, 626–638.CrossRefGoogle Scholar
Tort, M., Dubos, T. and Melvin, T. (2015), Energy-conserving finite-difference schemes for quasi-hydrostatic equations, Quart. J. Royal Meteorol. Soc. 141, 3056–3075.CrossRefGoogle Scholar
Walters, R. A. (2005), Coastal ocean models: Two useful finite element methods, Continent Shelf Res. 25, 775–793.CrossRefGoogle Scholar
Williamson, D. L., Drake, J. B., Hack, J. J., Jakob, R. and Swarztrauber, P. N. (1992), A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102, 211–224.CrossRefGoogle Scholar
Wimmer, G. A. and Tang, X. (2022), Structure preserving transport stabilized compatible finite element methods for magnetohydrodynamics. Available at arXiv:2210.02348.Google Scholar
Wimmer, G. A., Cotter, C. J. and Bauer, W. (2020), Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations, J. Comput. Phys. 401, 109016.CrossRefGoogle Scholar
Wimmer, G. A., Cotter, C. J. and Bauer, W. (2021), Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction, SMAI J. Comput. Math. 7, 267–300.CrossRefGoogle Scholar
Wood, N., Staniforth, A., White, A., Allen, T., Diamantakis, M., Gross, M., Melvin, T., Smith, C., Vosper, S., Zerroukat, M. et al. (2014), An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations, Quart. J. Royal Meteorol. Soc. 140, 1505–1520.CrossRefGoogle Scholar
Yamazaki, H., Shipton, J., Cullen, M. J. P., Mitchell, L. and Cotter, C. J. (2017), Vertical slice modelling of nonlinear Eady waves using a compatible finite element method, J. Comput. Phys. 343, 130–149.CrossRefGoogle Scholar
Zängl, G., Reinert, D., Rpodas, P. and Baldauf, M. (2015), The ICON (ICOsahedral Non-hydrostatic) modelling framework of DWD and MPI-M: Description of the non-hydrostatic dynamical core, Quart. J. Royal Meteorol. Soc. 141, 563–579.CrossRefGoogle Scholar
Zeitlin, V. (2018), Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models, Oxford University Press.CrossRefGoogle Scholar